A Novel Multiobjective Finite Control Set Model Predictive Control for IPMSM Drive Fed by a Five-Level Cascaded H-Bridge Inverter

In this work, a novel multiobjective voltage-vector-based (VVB) finite control set (FCS) model predictive control (MPC) for a permanent magnet synchronous machine drive fed by a three-phase five-level (3P-5L) cascaded H-bridge multilevel inverter (MI) is proposed. This algorithm aims to overcome the main issues relative to MPC implementation detected in the scientific literature for electric drives fed by cascaded H-bridge MIs (CHBMI). In detail, the goals are the minimization of computational cost by reducing the number of required predictions, the minimization of the switching devices state transitions, i.e., the switching losses minimization, and the common mode voltage (CMV) reduction. These goals are fulfilled through an offline optimization process; thus, no additional terms and weighting factors to be tuned are required for the cost function. Experimental validations are presented to prove the effectiveness of the proposed approach. In detail, an accurate electric drive performance comparison, both in steady state and dynamic working conditions, is carried out when the proposed VVB MPC and the cell-by-cell-based MPC are adopted. As comparison tools, current and voltage total harmonic distortion, apparent switching frequency, CMV amplitude, and torque ripple are adopted.

required switching devices, with respect to other MI topologies, and fault-tolerant capability [1], [2].CHBMI has been widely adopted in high-voltage high-power grid-connected applications as static compensator (STATCOM) or power flow controller (PFC) [3].However, it has aroused considerable interest also in the field of medium-voltage industrial electrical drives, since it results to be the most efficient MI topology in the range of 4. 16-13.8kV, regardless of the switching devices technology [4], [5].Moreover, CHBMI represents an interesting solution for the e-mobility sector [6].In detail, it allows for easy integration with the battery pack and an increment of total dc link voltage without increasing the switching devices voltage stress [7], [8].
In order to maximize the performance of these systems in terms of power losses and control flexibility, the scientific literature efforts focus on new control software solutions such as the model predictive control (MPC).MPC is a control strategy which is gaining considerable interest in the last ten years, thanks to a strong technological advancement in the field of microelectronics, which led to an increment in the controllers computational capability [9].In detail, the finite control set (FCS) MPC allows for obtaining the optimal control action by solving a state-space model-based integer quadratic optimal control problem (IQOCP) over a prediction horizon [10], [11].The control objectives are synthesized into a cost function, which allows for fulfilling several conflicting goals.The adoption of the controlled system state space model allows for considering system nonlinearities, which are typically correlated with converter switching behavior in power electronics and electrical drive fields.
It is a challenging task solving the IQOCP into a sampling interval T s when MI is adopted, due to the higher number of provided voltage vectors (VVs) with respect to traditional two-level voltage source inverters (2L-VSIs), i.e., the future system state must be predicted several times depending on the number of available VVs.Moreover, when CHBMI is adopted, VVs and gate control signals (GCSs) sets redundancies entail a further increase of the control complexity.Several strategies have been proposed in the literature to reduce the MPC computational complexity for CHBMI-fed systems.In [12], a cell-by-cell-based (CBCB) FCS-MPC for a cascaded H-bridge rectifier (CHBR) is proposed.In detail, per each H-Bridge (HB) phase module, the IQOCP is independently formulated and solved.In [13], a hierarchical FCS-MPC is proposed for a grid-connected single-phase CHBMI.In detail, the candidate GCSs set is selected over a subregion of a 2-D control plane.The control plane allows for the candidate switching state set preselection and additional cost function terms can be deleted.In [14], an FCS-MPC for a CHBR with available VV set online variation is formulated.The VVs set is defined depending on the system working conditions (i.e., transients or stationary conditions, faults conditions, and asymmetric loads).In [15], the FCS-MPC computational growth reduction from exponential form to polynomial form is addressed by the authors.In detail, the IQOCP is split into two suboptimization problems, which deal with the current control and the optimal GCSs set selection, respectively.Zhang et al. [16] authors improved the previous algorithm with a simplified branch and bound (BnB) approach.In [17], modulated model predictive control (M 2 PC) for a CHB-based synchronous compensator is proposed with voltage self-balancing ability.
In [18], a fascinating FCS-MPC with reduced available VVs candidates is proposed: in detail, the VVs candidate set at the current instant is composed of adjacent VVs to the previous optimal VV.Redundant VVs are eliminated to minimize the common mode voltage (CMV) which is a prominent feature in electrical drive applications since high CMV produces bearing currents that determine an early degradation of the bearing and the winding insulation [19].Several researchers faced the problem by adopting the MPC due to its capability to fulfill different conflicting control goals [20].Parvathy and Kumar [21] and Guo et al. [22] propose FCS-MPC-based strategies for CMV minimization in traditional 2L-VSI-based electrical drives: in detail, to avoid the use of zero voltage vectors (ZVV), which determine the highest CMVs, some adjacent and nonadjacent VVs are applied for a specific time duration over a sampling period, to guarantee the lowest voltage ripple.Nevertheless, the application of nonadjacent VVs causes high voltage dv/dt, which are correlated with the switching devices voltage stress and electromagnetic interferences.In [23], MPC is adopted for CMV suppression for a linear permanent magnet synchronous motor (LPMSM) drive fed by a neutral point clamped inverter (NPCI) where redundant VVs with higher CMVs are discarded.Wang et al. [24] exploit the FCS-MPC to dynamically adjust the number of candidate vectors according to the predicted current ripple.In detail, it emerges that the ZVV must be used only in the low-speed range.
Looking at the scientific literature, the following considerations must be highlighted.
1) The majority of the works in the literature that deal with FCS-MPC with CHBMI are designed for gridconnected applications, where it usually operates as a rectifier.Moreover, most of the discussed control strategies require additional computation stages to evaluate the current system state, to define the VVs candidate set online, and its application time duration.2) Most of the papers that deal with CMV minimization take into account traditional 2L-VSIs, additional terms are introduced into the cost function and this makes the weighing factor tuning an annoying activity; 3) Looking at [18], although both the computational cost reduction and CMV minimization goals are fulfilled, several issues are left unanswered: CHBMI exhibits not only VVs but also GCSs redundancy, i.e., the same phase voltage sets can be synthesized by several switching state combinations; managing GCSs redundancies is a challenging task.Moreover, the FCS-MPC has been designed on a passive RL load, and its impact on an electrical drive is missing.The practical algorithm implementation is not discussed.This work aims to overcome the main weaknesses of the control strategy presented in [18] by designing and implementing a novel multi-objective voltage-vector-based (VVB) FCS-MPC for PMSM drives fed by CHBMI.In detail, the control must fulfill the following goals: computational burden minimization, CMVs minimization, GCS transitions minimization, and phase voltages dv/dt minimization.It must be underlined that the control optimization process is carried out offline, to reduce the online computational amount and, once it is completed, the proposed goals are automatically fulfilled, i.e., no additional cost function terms are required, unlike other proposed FCS-MPCs in the literature.This feature allows for avoiding the weighting factors tuning process, which is an annoying activity when the cost function is composed of many terms.The offline optimization process deals with the redundant VVs with high CMVs elimination and optimal GCS set selection, to minimize the switching devices state transitions.An online selection algorithm is implemented to predict the future system state only with respect to the adjacent VVs to the currently applied one.This constraint allows for minimizing both the optimal control problem computational burden since only seven predictions must be carried out at each sampling period and minimizing the voltage dv/dt since phase voltage instantaneous excursion is limited to one HB dc link voltage.
For experimental validation purposes, a test bench with an interior permanent magnet synchronous motor (IPMSM) drive fed by a three-phase five-level (3P-5L) CHBMI has been set up.Experimental tests are carried out to verify the proposed control effectiveness: in detail, the electric drive performance, both in steady state and dynamic working condition, are analyzed and compared when the proposed VVB-FCS and the CBCB-MPC presented in [12] are adopted.It must be underlined that the algorithm presented in [12] has been chosen as the comparison target for the following reasons: first, the CBCB approach allows for reducing the control computational complexity by introducing some specific switching constraints that exploit the CHB topology; this constraints-based control design approach is homogeneous with the one adopted in this work; moreover, no BnB approach must be adopted to solve the optimal control problem, and this makes the control implementation quite easy; last but not least, although the CBCB algorithm is proposed for a single-phase CHB rectifier, its strong correlation with the converter topology allows to easily adapt the algorithm to a three-phase electric drive application.All these features make the CBCB algorithm a direct competitor of the VVB MPC.As comparison tools, current and voltage total harmonic distortion, apparent switching frequency, CMV amplitude, and torque ripple are adopted.This work is organized as follows.Section II describes the FCS-MPC mathematical formulation.Section III presents the proposed algorithm.Section IV describes the test bench; Section V presents and analyzes the experimental results.Finally, Section VI summarizes the work results.

II. MODEL PREDICTIVE CONTROL FORMULATION
In this section, the IQOCP is formulated.In detail, both IPMSM and the 3P-5L CHBMI control models are introduced.The electric drive control scheme is reported in Fig. 1.
In detail, motor speed and currents are chosen as controlled variables.The outer control loop allows generating the current references in the d-q reference frame, for this purpose, the proportional and integral (PI) controller is adopted.The inner control loop is synthesized as an FCS-MPC.

A. IPMSM control model
For control formulation purposes, the IPMSM continuoustime state-space model in the two-phase d-q reference frame is considered with where v dq is the stator VV, i dq is the stator current vector, L d and L q are the direct and quadrature inductances, R is the stator winding resistance, ω m is the mechanical rotor speed, λ PM is the permanent magnet flux linkage, and p is the pole pairs of the machine.By discretizing equations ( 1) with forward-Euler approximation, the discrete-time statespace model is obtained with

B. CHBMI Control Model
The control model of the 3P-5L CHBMI, whose circuit diagram is reported in Fig. 2, must be defined, and integrated with the motor model.
With this aim, the input VV v dq must be expressed regarding the converter switching states.In detail, with respect to the circuit diagram reported in Fig. 2, and taking into account that, in order to avoid leg short circuit, switching devices on the same HB leg must work in a complementary way, the phase voltage can be expressed as follows: where v dc is the HB dc-link voltage and S j,x y is the HB leg state variable, j ∈ {A, B, C} identifies the converter phase, x ∈ {1, 2} identifies the HB phase module, and y ∈ {1, 4} identifies the HB leg.Phase VVs in d-q reference frame and ABC reference frame are linked by the relation with where θ is the rotor angular position.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

C. Cost Function
A prediction horizon N p = 1 is chosen and the cost function J is formulated as follows: where i * dq is the reference currents vector and i dq is the future state current vector.The cost function penalizes the input current error only.

III. VOLTAGE VECTORS-BASED OPTIMIZATION PROCESS
In this section, the adopted strategy to simplify the IQOCP resolution is discussed.In detail, the offline optimization process and the simplified control algorithm implementation are discussed.

A. Offline Optimization Process
In Fig. 3, the VVs of a 3P-5L-CHBMI in the α-β reference frame are represented.In order to minimize the computational burden, the number of predictions must be minimized.With this aim, the set of available VVs at the kth sampling period consists only of the adjacent VVs to the currently applied one.As a way of example, with respect to Fig. 3, at the kth sampling period, the applied VV is v 16 , which is identified by the red circle.
At the k + 1th sampling instant, the system can move to the adjacent vectors v 17 , v 24 , v 23 , v 15 , v 9 , and v 10 or remain in v 16 .In general, the number of available VVs to be tested in a sampling interval is equal to 7 (i.e., six adjacent vectors and the currently applied vector itself), except when the current vector belongs to the outer hexagon; in this case, the number of available input variable sets is equal to 4 when the current vector lies on one of the six edges of the hexagon, and 5 in other cases.The total number of VVs in the α-β reference frame is obtained as follows: where n l is the number of phase voltage levels and m is the number of phases.Looking at Fig. 3, it can be noted that most of the 125 VVs are superimposed, which means that most of the VVs are redundant.In detail, by identifying four concentric hexagons and by moving from the outer to the inner one, the degree of redundancy assigned to the VVs which lie on each hexagon goes from 0 (i.e., no redundant VVs) to 3. The ZVV has a degree of redundancy equal to 4. In order to minimize the CMVs, per each VV, the phase voltage combinations (PVCs) which minimized the CMV are saved and other combinations are discarded.CMVs are computed according to the following relation: As a way of example, the vector v 7 is considered: it can be obtained by the following PVCs, expressed in the sequence of the phases A B C and in per unit (p.u.) with respect to the voltage v dc : (−1 2 −1) and (−2 1 −2).These PVCs generate CMVs, expressed in p.u., of 0 and −1 V respectively.In this case, the PVC is saved, and the last is discarded.About vector v 14 , it can be obtained by the PVCs (0 2 0), (−1 1 −1), and (−2 0 −2) which generate CMVs of 2/3, −1/3, and −4/3 V, respectively.In this case, the PVC (−1 1 −1) is saved and the others are discarded.
Going on with the remaining VVs, it can be noted that there is always only one PVC which minimizes the CMV, this makes the VVs selection very easy.In Fig. 4(a) and (b), the PVCs which minimize the CMVs per each VV and the corresponding CMVs are reported, respectively.Now, the GCSs are considered.According to (5), the VVs can be synthesized by a number of GCSs combinations N GCSs expressed as follows: which is equal to 4096 when n l = 5 and m = 3.Therefore, the nonredundant VVs selected in the previous step can be synthesized by several GCSs sets.In order to minimize the GCSs transitions, which are correlated with the switching losses, one optimal GCSs set must be assigned to each VV.The GCSs assignment procedure is a trial-and-error iterative process, which is briefly discussed below.Taking into account the generic state transition v i ➜ v j and assuming that the GCSs set S i (expressed as a row of 12 Boolean elements) is always known since it has been assigned previously, the goal is to find the set S j which minimizes the transition S i ➜ S j , for every v i adjacent to v j .As a way of example, if the VV v 1 is currently applied, the system can move to vectors v 2 , v 6 , and v 7 .Assuming that S 1 is known, in Table I, the transition v 1 ➜ v 6 is considered: it can be noted that the number of transitions is the same and equal to 1 for every S 6 .Therefore, the optimal set is chosen arbitrarily (green row in Table I).The same considerations can be applied to the transition v 1 ➜ v 7 , which is summarized in Table II.Now, transition v 6 ➜ v 7 is considered.In detail, the goal is to verify if the previously selected sets S 6 and S 7 allow also minimizing transitions v 6 ➜ v 7 , which is summarized in Table III.Looking at Table III, the number of transitions is equal to 1 and 3, depending on the considered GCS set.Sets that determine three state transitions (red rows in Table III) are discarded.Among the remaining sets, the first one (green row in Table III) Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.coincides with the previously selected set.As can be seen, the selected sets S 6 and S 7 minimize also the transitions v 6 ➜ v 7 .
Going on with all the remaining VVs transitions, one GCS set is assigned to each VV.The optimal GCSs are reported in Fig. 4(c).
Once a unique match between VVs, PVCs, and GCSs sets is found, the offline optimization process is completed.For control algorithm implementation purposes, the selected sets of PVCs and GCSs are stored in two look-up tables (LUTs), called LUT-1 and LUT-2, respectively.This approach allows to easily define the set of available PVCs and relative GCSs at the sampling instant k, depending on the previous optimal control action, without affecting the algorithm computational cost, since the access to the elements of LUTs is instantaneous and does not require extra-computations.This aspect is remarked in Section III-B, where the algorithm implementation is discussed.The final optimal control problem is formulated as follows: where U (k) identifies the available input variables set at the kth sampling instant, which depends on the VV that is currently applied to the system.

B. Control Algorithm Implementation
In this section, the algorithm implementation is discussed.The proposed control algorithm is summarized in Algorithm 1. First, operations that can be executed once per sampling period, i.e., the delay time compensation, are implemented.In detail, due to the fact that the control execution time is always higher than 0, a delay is introduced between the sampling instant and the control action application instant.Such a delay depends on the algorithm computational complexity and the adopted controller.In this application, the control action is instantaneously applied when it is available, instead of being stored and applied to the next sampling instant, as is usual when control with a modulator is adopted.As a consequence, the delay time T d coincides with the execution time.These aspects are illustrated in Fig. 5: in detail, the ideal case with a null execution time (i.e., a delay time T d = 0) and the real case with a nonnull execution time (i.e., a delay time T d ̸ = 0) are reported in Fig. 5(a) and (b), respectively.Ignoring the delay time in practical applications can determine nonaccurate future system state predictions, which lead to the application of suboptimal solutions to the controlled system and, as a consequence, current and torque ripple [25].
In order to take into account such execution time, a delay compensation strategy is adopted.By assuming that the delay time T d is known and constant for each sampling period, and assuming the speed ω m and the angular position θ constant over the sampling period, an initial prediction is performed to project the system state from instant k to the instant k + ε, where ε is the control action application instant, which is between the kth and k + 1th sampling instants and is correlated with the delay time T d , according to Fig. 5.The system state at the instant k +ε represents the new current state to adopt for the future state predictions.The initial prediction is performed according to the following equation: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
with A d , B d , and K d being as in (4) but replacing the delay time T d with the sampling time T s , as follows: and v dq,opt is the optimal input vector applied to the system at the instant k + ε − 1, which is also applied at the sampling instant k and whose value is known.Once the system state at the instant k + ε is estimated, it is possible to predict the state at the instant k + ε + 1, that is, according to Fig. 5, the future control action application instant, as follows: with matrices A, B, and K being as in (4).Replacing ( 6) and ( 13) in ( 15) and rearranging, the following relation is Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Algorithm 1 Model Predictive Control 1: Compute i dq, p1 (k + ε + 1), according to (17) 3: Compute BT(k) 4: Access to LUT1: Define the set of 7 v N (k + ε) and v x , based on v x,opt (k + ε − 1) 5: Initialize cost function J 0 6: for y = 1: 7 7: Compute i dq (k + ε + 1, y) according to (18) 8: Compute J (y), according to (8) 9: if J (y) ≤ J 0 10: end if 13: end for 14: Access to LUT2: Define S opt (k + ε), based on v x,opt (k + ε) 15: Apply S opt (k + ε) to the system 16: end function Looking at (16), it can be noted that only the term Bv dq (k + ε) depends on the phase VV v N .Therefore, the term i dq, p1 (k + ε + 1) can be defined This term can be computed only once per sampling period since it is the same for all seven predictions.Thus, ( 16) can be rewritten as follows: Next, based on the currently applied VV, the set of available VVs and the corresponding PVCs is defined.This operation is carried out by acceding to an LUT: in detail, each VV is identified by a number v x , according to the enumeration in Fig. 3, which represents the address to accede to the LUT, and a PVC, according to Fig. 4(a).Since the currently applied VV is known, the set of available seven VVs for future state predictions is identified.It must be underlined that this operation is extremely fast because access to LUTs does not affect the computational burden and the algorithm execution time.Once the seven VVs are known, the seven corresponding predictions are carried out.The VV which minimizes the cost function J represents the solution to the IQOCP, and the corresponding GCS set is identified through another LUT and applied to the system.

C. Computational Cost Comparison With CBCB-MPC
As discussed in the introduction of this work, the CBCB-MPC algorithm presented in [12] represents a computationally cost-effective, easy-to-implement algorithm that exploits the potentialities of the CHB topology; therefore, it becomes the best available comparison target of this work.To make the comparative analysis as fair as possible, the CBCB-MPC implementation has been out almost in the way as discussed in Section III-B about the MPC: in detail, the electric drive control model, the cost function, the delay time compensation the model parameters, and the sampling frequency are the same as discussed in (3)-( 8) and ( 13) and in Table IV, respectively.To minimize the computational cost, also in this case the prediction equation has been split into a term that can be computed only once per sampling period and a term, which depends on the available phase voltages at the current sampling instant, which must be computed several times to carry out the future state predictions, according to (17) and (18).According to [12], the algorithm for the single-phase rectifier must carry out two or three predictions, depending on the current inverter state.To adapt the algorithm for a three-phase electrical drive, the PVCs must be taken into account.Thus, the maximum number of predictions becomes 3 3  = 27.To carry out these predictions, the second term of ( 18) is computed into three nested for loops, one per each phase.
The adopted controller for this application is the system on module (SoM) sbRIO 9651, which consists of an field-programmable gate array (FPGA) and a DSP module and they can be programed independently in the LabVIEW environment.The presented algorithm is entirely implemented on the FPGA target with single-precision floating point data.The real-time target has been adopted only for the implementation of the graphical user interface (GUI).In Table V, the control parameters adopted for the CBCB-MPC and the VVB-MPC are reported: in detail, the adopted sampling period is equal to 100 µs in both cases, the computational time has been experimentally measured and it is equal to 55 and 23 µs for CBCB-MPC and VVB-MPC, respectively.It can be noted that CBCB-MPC requires more than twice the time needed by the VVB-MPC to elaborate the control action, due to the higher number of predictions required.In this case, the VVB approach results in being more efficient than the phase-by-phase-based one.In Table V, the control computational resources employed for the CBCB-MPC and the VVB-MPC are reported.It must be underlined that employed computational resources and computational time deal with the whole control, which consists not only of the current control, which is synthesized as the proposed FCS-MPC but also includes the sampling process, the angle tracking observer (ATO) algorithm for the speed and position measurement, the outer speed loop, which is synthesized as common PI controller.These parts of the control are deeply discussed in [26] and [27].

IV. TEST BENCH
In this section, the test bench, whose photograh is reported in Fig. 6, is discussed.The electrical drive consists of six MOSFET-based power HBs powered by six dc power supply RSO-2400 and six poles three-phase BLQ-40 IPMSM.Electrical drive technical data are summarized in [27,.A MAGTROL HD-715 hysteresis brake controlled by a MAGTROL DSP6001 high-speed programmable dynamometer is used to apply a load torque to the IPMSM.The electrical quantities are acquired by Teledyne LeCroy MDA 8038HD oscilloscope, equipped with three high voltage differential probes Teledyne Lecroy HVD3106A 1 kV, 120 MHz, and three high sensitivity current probe Teledyne Lecroy CP030A ac/dc, 30 A rms, and 50 MHz.

V. EXPERIMENTAL RESULTS
In this section, experimental results are presented to validate the effectiveness of the proposed control algorithm.In detail, an extensive comparative analysis is carried out with the CBCB-MPC presented in [12].For steady-state performance analysis, a set of working points (WPs) have been defined as a function of IPMSM working conditions in terms of speed n and load torque T. considered are summarized in Table VI.For each identified WP, phase voltages and phase currents are measured with Teledyne LeCroy MDA 8028HD oscilloscope, with an observation window of 1 s and a sampling frequency of 1 MS/s.To investigate the electromagnetic torque behavior, for each identified WP, the electromagnetic torque T em is estimated starting by the d-q control currents acquired.The currents are sampled with a sampling frequency of 10 kHz, which is the same adopted for the control algorithm, with an observation window of 1 s.As performance metrics, the switching frequency which is correlated with the converter switching losses, the current and voltage THD which are correlated with the copper losses and the iron losses, respectively, the CMV and the torque ripple are considered.
With respect to the dynamic behavior analysis, a cycle composed of a 0-3000 r/min acceleration, including no-load and rated-load operations, has been carried out.During dynamic analysis, the maximum allowed transitory currents were set to i q = 10 A and i d = 0, respectively.IPMSM phase voltages, phase currents, load torque applied by the brake, and speed signals are acquired with an MDA 8028HD oscilloscope by setting an observation window of 5 s and a sampling frequency of 500 kS/s.

A. Steady-State Condition Analysis
The phase voltages produced by the CBCB-MPC and the VVB-MPC in the time and frequency domain are reported in Fig. 7(a)-(u), respectively, when the load torque is fixed equal to 1.8 N•m and the speed is varied over the defined working range.Concerning the VVB-MPC, phase voltages exhibit three voltage levels when the speed is lower than 2000 r/min, and five levels when the speed is higher than 2000 r/min.Phase voltages with CBCB-MPC are characterized by a more chaotic behavior and exhibit the fifth level even in the low-speed working range.However, in both cases, phase voltages are characterized by dv/dt at most equal to one HB dc link voltage over the entire working range.Voltage spectra with VVB-MPC present harmonics lower than 5% of the fundamental except when the speed is equal to 4000 r/min when the overmodulation region is reached.Concerning the CBCB-MPC, voltage spectra are characterized Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.A comparison between instantaneous CMV trends is reported in Fig. 7(v)-(z): with respect to VVB-MPC, the CMVs maximum instantaneous value is equal to 18.33 V, except when the speed is equal to 4000 r/min, where it is equal to 36.66 V; a much worse behavior can be noted with the CBCB-MPC, where the most frequent peak value is equal to 55 V and the maximum peak value is equal to 110 V which occur in the low-speed range.This behavior is emphasized in Fig. 8, where CMV instantaneous peak values over the defined working range are reported.It can be noted that the control goal of minimizing the CMV with the VVB-MPC, which is synthesized in Fig. 4(b), has been reached over the entire defined working range, since the CMV peak is always equal to v dc /3 (i.e., 18.33 V, when v dc = 55 V), except in WPs 1, 6, and 11, where the CMV peak reaches the value of 2v dc /3 equal to 36.66 V.This behavior is justified by taking into account that only when the speed is equal to the rated value and the load is higher than half the rated torque, it is mandatory to employ the VVs v 1 , v 5 , v 27 , v 35 , v 57 , and v 61 are employed where a higher CMV peak value cannot be avoided.On the other hand, no attention is paid to CMVs in the CBCB-MPC algorithm design.Only when the speed reaches the rated value, CMV behaviors become similar.
The phase currents in the time and frequency domain, when the CBCB-MPC and the VVB-MPC are adopted, are shown in Fig. 9(a)-(p), respectively, when the load torque is varied over the defined working range and the speed is constant and equal to 3000 r/min.Comparing CBCB-MPC and VVB-MPC current spectra, it can be noted that also, in this case, VVB-MPC guarantees a lower harmonic content over the entire working range, although the difference is less marked, if compared with voltage spectra: the worst case takes place when speed is equal to 200 r/min and the low-frequency harmonics reach the maximum value of 4% and 3% when CBCB and VVB are considered, respectively.
In Fig. 10, a half-period of the phase voltages expressed in p.u. and the GCSs related to the VVB-MPC are reported when the load torque is set to 0 and the speed is 3000 r/min.It is easy to see that the GCS constraints are fulfilled, since only one or, at least, two GCSs change state over two adjacent sampling instants, according to Fig. 4(c).The apparent switching frequency f sw over the defined working range for the CBCB-MPC and the VVB-MPC are reported in Fig. 11(a) and (e), respectively.Switching frequency estimation has been carried out by counting the phase voltage level transitions N t over the observation window and dividing by double the observation window T w , according to (19) It can be noted that, in the considered working range, the apparent switching frequency varies from 1800 to 2600 Hz with an average apparent switching frequency equal to 2200 Hz when CBCB-MPC is adopted.Instead, the apparent switching frequency varies from 1400 to 1900 Hz with an average apparent switching frequency equal to 1700 Hz when VVB-MPC is adopted.It must be underlined that VVB-MPC guarantees at the same time a lower switching frequency and a better voltage and current harmonic content over the entire working range if compared to the CBCB-MPC.This behavior is confirmed by the current and voltage THD maps over the adopted working range, reported in Fig. 11(b) and (c) and (f) and (g).In detail, it can be noted that the current THD varies in the range of 7%-25% and 5%-21% when CBCB-MPC and VVB-MPC are adopted, respectively.In both cases, for a fixed speed, the THD drastically decreases when the load torque increases.This behavior is correlated with an increment of the fundamental frequency.Moreover, it can be noted a correlation between the current THD and the speed, such that the Total harmonic distortion (THD) slowly increases when the speed increases.This behavior can be justified by a progressive decrement in the switching frequency when the speed increases.About the voltage THD, it varies in the range of 40%-300% and 30%-200%, respectively.It can be noted that the THD variation is strictly correlated with the speed.In detail, when the speed increases, the THD decreases.Moreover, the voltage THD is quite independent of the load torque.
The CMV rms value over the defined working range is reported in Fig. 11(d) and (h), for CBCB and VVB-MPC, respectively.In detail, it can be noted that the CMV rms value over the defined working range is always quite low and the voltage variation over the working range is quite small, 12-18 V, when VVB-MPC is adopted.This analysis confirms that the goal of minimizing the CMV with VVB-MPC has been fulfilled.Moreover, the CMV harmonic content results to be quite independent of the working range.Concerning CBCB-MPC, voltage variation over the working range is 22-32 V, thus, CMV rms values are higher than VVB-MPC over the entire working range, as was expected.
To investigate the electromagnetic torque trend and the torque ripple, the electromagnetic torque has been estimated starting with the control current in the d-q reference frame, according to the following relation: A comparison between the electromagnetic torque trends generated with CBCB-MPC and VVB-MPC when the load torque varies over the defined range and when the speed is equal to 2000 and 3000 r/min is reported in Fig. 12.
It can be noted that the VVB-MPC guarantees a lower torque ripple in every working condition, except when the speed is 3000 r/min and the torque is 1.35 N•m, where torque ripple values are comparable.To quantify the torque ripple Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.over the entire defined working range, the torque ripple rms value, expressed in percent with respect to the average torque, has been adopted according to [28] T where N s is the number of samples over the defined observation window and T em,avg is the average electromagnetic torque value.In Fig. 13(a) and (b), the torque ripple maps over the defined working range with the two compared strategies are reported.
It can be noted that the torque ripple varies in the range of 2%-24% and 1%-16% when CBCB-MPC and VVB-MPC are adopted, respectively.In detail, both torque ripple map trends are very similar to the respective current THD% trends, such that there is a strong correlation with the motor speed.Also in this case, VVB-MPC guarantees a lower torque ripple over the entire working range.

B. Dynamic Condition Analysis
In Fig. 14, a comparison between the electric drive dynamic behavior obtained with the two compared strategies is reported.In detail, IPMSM phase voltages, phase currents, speed, and load torque trends during the whole dynamic cycle are reported in Fig. 14(a) and (b), respectively.It is interesting to note that, in both cases, the currents reach the reference values in about 2 ms with a negligible overshoot.The speed transient ends in about 500 ms with about 200-r/min overshoot.The system exhibits also a good rejection of external disturbances, as can be seen when the load torque is applied and next removed.In detail, the speed transient due to the external disturbance ends in about 0.3 s.The reported Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.trends state that both algorithms guarantee good dynamic performance, with a reduced speed and torque ripple.Thus no remarkable differences in the dynamic response have to be underlined.

VI. CONCLUSION
In this work, the design and implementation of a novel multiobjective VVB FCS-MPC for PMSM drive fed by a 3P-5L CHBMI have been carried out.The control goals deal with computational burden minimization, CMVs minimization, GCSs transitions minimization, and phase voltages dv/dt minimization.The first goal is fulfilled by imposing that, at a certain sampling instant, future state prediction must be carried out by considering the set of adjacent VVs with respect to the currently applied VV.In this way, only seven (instead of 4096) predictions must be carried out per sampling period.This constraint also allows for fulfilling the voltages dv/dt minimization.
The second goal is fulfilled by the elimination of redundant VVs which do not guarantee the minimum CMVs.The third goal is fulfilled by selecting GCSs such that only one or, at most, two, GCSs change state in a transition, i.e., only one or two HBs legs are changing state.These goals are reached through an offline optimization process.Thanks to the offline optimization procedure, the proposed online FCS-MPC algorithm does not require additional computation stages if compared with a nonconstrained FCS-MPC.Indeed, the online VVs selection is carried out by acceding to LUTs; this operation does not affect the computational burden and the execution time.Moreover, the offline optimization process allows for reaching the proposed goals without synthesizing them into the cost function.Thus, no additional terms must be added to the cost function, with respect to a standard FCS-MPC and no weighting factors must be tuned.For control validation purposes, a detailed experimental comparative analysis with a CBCB-MPC strategy both in steady-state and dynamic working conditions has been carried out.Experimental results confirm that the proposed algorithm meets the proposed control goals and, at the same time, it guarantees better electric drive performance in steady-state working conditions in terms of voltage and current harmonic content, voltage and current THD, switching frequency, and torque ripple compared to the CBCB-MPC.Thus, the proposed strategy allows to reduce at the same time the switching frequency, which is correlated with the switching losses, and the torque ripple.The system's dynamic behavior is quite fast, and the system is able to reject external disturbance and work in the defined set point.Comparison with the CBCB-MPC shows no sensitive difference in terms of dynamic performance.

Fig. 4 .
Fig. 4. (a) Optimal PVCs for every VV in p.u.(b) CMV values per each VV in p.u. (c) GCSs which minimize the switching transitions reported on the α-β plane.

Fig. 8 .
Fig. 8.Comparison between peak values of the CMV in each WP with CBCB-MPC and VVB-MPC.

TABLE V EMPLOYED
COMPUTATIONAL RESOURCES ON SOM SBRIO 9651 Fig. 6.Test bench.

TABLE VI ELECTRIC
DRIVE WORKING POINTS